I tend to use the method that most people dislike, but it works well for me. I use the lasso select tool to loosely select the object I want to cut out. I copy it and paste it onto a new background. Then I zoom in, a lot, and use the lasso tool with feathering set to zero, to cut away everything that isn't what I want.. Then, still zoomed in, I use the blur tool to soften the cut edges. Cut and pasted images just don't look right with those very sharp edges. You don't blur a lot, just enough to blend it in a little. I've been doing it that way since Photoshop 3, and I doubt I'd be comfortable or competent doing it any other way. It's a lot of work, but it works for me.

Talent is God's gift to you. Using it is your gift to God.